The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain

We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F（t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Granata, A. (2015) The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain. Advances in Pure Mathematics, 5, 100-119. doi: 10.4236/apm.2015.52013.

 [1] Granata, A. (2007) Polynomial Asymptotic Expansions in the Real Domain: The Geometric, the Factorizational, and the Stabilization Approaches. Analysis Mathematica, 33, 161-198. http://dx.doi.org/10.1007/s10476-007-0301-0 [2] Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part I: Unsatisfactory or Partial Results by Classical Approaches. Analysis Mathematica, 36, 85-112. http://dx.doi.org/10.1007/s10476-010-0201-6 [3] Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part II: Factorizational Theory. Analysis Mathematica, 36, 173-218. http://dx.doi.org/10.1007/s10476-010-0301-3 [4] Granata, A. (2011) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part I: Two-Term Expansions of Differentiable Functions. Analysis Mathematica, 37, 245-287. (For an Enlarged Version with Corrected Misprints see: arxiv.org/abs/1405.6745v1 [mathCA]. http://dx.doi.org/10.1007/s10476-011-0402-7 [5] Granata, A. (2014) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II: The Factorizational Theory for Chebyshev Asymptotic Scales. Electronically Archived—arXiv: 1406.4321v2 [math.CA]. [6] Granata, A. (2015) The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results. Advances in Pure Mathematics, 5, 1-20. http://dx.doi.org/10.4236/apm.2015.51001 [7] Haupt, O. (1922) über Asymptoten ebener Kurven. Journal für die Reine und Angewandte Mathematik, 152, 6-10; ibidem, 239. [8] Sanders, J.A. and Verhulst, F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York. [9] Corduneanu, C. (1968) Almost Periodic Functions. Interscience Publishers, New York. [10] Faedo, S. (1946) Il Teorema di Fuchs per le Equazioni Differenziali Lineari a Coefficienti non Analitici e Proprietà Asintotiche delle Soluzioni. Annali di Matematica Pura ed Applicata (the 4th Series), 25, 111-133. http://dx.doi.org/10.1007/BF02418080 [11] Hallam, T.G. (1967) Asymptotic Behavior of the Solutions of a Nonhomogeneous Singular Equation. Journal of Differential Equations, 3, 135-152. http://dx.doi.org/10.1016/0022-0396(67)90011-3 [12] Hukuhara, M. (1934) Sur les Points Singuliers des équations Différentielles Linéaires; Domaine Réel. Journal of the Faculty of Science, Hokkaido University, Ser. I, 2, 13-88. [13] Ostrowski, A.M. (1951) Note on an Infinite Integral. Duke Mathematical Journal, 18, 355-359. http://dx.doi.org/10.1215/S0012-7094-51-01826-1 [14] Agnew, R.P. (1942) Limits of Integrals. Duke Mathematical Journal, 9, 10-19. http://dx.doi.org/10.1215/S0012-7094-42-00902-5 [15] Hardy, G.H. (1911) Fourier’s Double Integral and the Theory of Divergent Integrals. Transactions of the Cambridge Philosophical Society, 21, 427-451. [16] Hardy, G.H. (1949) Divergent Series. Oxford University Press, Oxford. (Reprinted in 1973) [17] Blinov, I.N. (1983) Absence of Exact Mean Values for Certain Bounded Functions. Izvestija Akademii Nauk SSSR. Serija Mathematicheskaja (Moscow), 47, 1162-1181. [18] Ditkine, V. and Proudnikov, A. (1979) Calcul Opérationnel. éditions Mir, Moscou. [19] Baumgartel, H. and Wollenberg, M. (1983) Mathematical Scattering Theory. Birkhauser Verlag, Berlin. [20] Ostrowski, A.M. (1976) On Cauchy-Frullani Integrals. Commentarii Mathematici Helvetici, 51, 57-91. http://dx.doi.org/10.1007/BF02568143 [21] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511721434 [22] Hartman, Ph. (1952) On Non-Oscillatory Linear Differential Equations of Second Order. American Journal of Mathematics, 74, 389-400. http://dx.doi.org/10.2307/2372004 [23] Hartman, Ph. (1982) Ordinary Differential Equations. 2nd Edition, Birkhauser, Boston. [24] Giblin, P.J. (1972) What Is an Asymptote? The Mathematical Gazette, 56, 274-284. http://dx.doi.org/10.2307/3617830